Contractions of the actions of reductive algebraic groups in arbitrary characteristic (Q1187480)

Let \(k\) be an algebraically closed field, and \(A\) a commutative \(k\)- algebra. Let \(G\) be a reductive algebraic group which acts rationally on \(A\), and \(B=TU\) a Borel subgroup of \(G\), and \(B^ -=TU^ -\) the Borel subgroup of \(G\) opposite to \(B\). Denote by \(A^ U\) the set of fixed points of \(A\) under the action of \(U\). The author of this paper relates properties of \(A\) to those of \(A^ U\). The author also constructs a \(G\)- action on \(R=(A^ U \otimes_ k k[G/U^ -])^ T\), and shows that there is a graded algebra \(\text{gr }A\) and an injective \(G\)-equivariant algebra homomorphism \(\Phi:\text{gr }A\to R\). Moreover, the action of \(G\) on \(\text{Spec}(A)\) is a flat deformation of the action of \(G\) on \(\text{Spec(gr }A)\), and \(\Phi\) is an isomorphism if and only if \(A\) has a good filtration.